ON PERIODIC PROBLEMS FOR THE NONLOCAL
POISSON EQUATION IN THE CIRCLE
Batirkhan Turmetov1, Maira Koshanova2, Moldir Muratbekova3
1,2,3Khoja Akhmet Yassawi International
Kazakh-Turkish University
Turkistan - 161200, KAZAKHSTAN

Abstract

This work is devoted to the study of the methods of solving the boundary value problem with transformed arguments. In the studied problems, the arguments are transformed by mapping the type of involution. Moreover, these mappings are used both in the equation and in the boundary conditions. The equation under consideration is a nonlocal analogue of the Poisson equation. The boundary conditions are specified as a relation between the value of the desired function in the upper semicircle and its value in the lower semicircle. Two types of boundary conditions are considered. They generalize the known periodic and antiperiodic conditions for circular regions. When solving the main problems for the classical Poisson equation, auxiliary problems are obtained. Using well-known assertions for these auxiliary problems, theorems on the existence and uniqueness of solutions are proved. Exact conditions for the solvability of the studied problems are found.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 36
Issue: 5
Year: 2023

DOI: 10.12732/ijam.v36i5.11

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References

  1. [1] A. Ashyralyev, A.M. Sarsenbi, Well-posedness of an elliptic equation with involution, Electron. J. Differential Equations, 284 (2015), 1-8.
  2. [2] L.C. Evans, Partial Differential Equations, 2nd Edition, Graduate Studies in Mathematics, V19 AMS (2010).
  3. [3] T.Sh. Kal’menov, U.A. Iskakova, A criterion for the strong solvability of the mixed Cauchy problem for the Laplace equation, Dokl. Math., 75 (2007), 370-373; https://doi.org/10.1134/S1064562407030118.
  4. [4] T.Sh. Kal’menov, B.T. Torebek, A method for solving ill-posed nonlocal problem for the elliptic equation with data on the whole boundary, J. Pseudo-Differ. Oper. Appl., 10 (2019), 177-185; https://doi.org/10.1007/s11868-017-0231-y.
  5. [5] V.V. Karachik, A.M. Sarsenbi, B.Kh. Turmetov, On the solvability of the main boundary value problems for a nonlocal Poisson equation, Turk. J. Math., 43 (2019), 1604-1625; https://doi.org/10.3906/mat-1901-71.
  6. [6] V.V. Karachik, B.Kh. Turmetov, Solvability of one nonlocal Dirichlet problem for the Poisson equation, Novi Sad J. Math., 50 (2020), 67-88; https://doi.org/10.30755/NSJOM.08942.
  7. [7] M. Koshanova, M. Muratbekova, B. Turmetov, On periodic boundary value problems with an oblique derivative for a second order elliptic equation, International Journal of Applied Mathematics, 34, No 2 (2021), 259-271; http://dx.doi.org/10.12732/ijam.v34i2.4.
  8. [8] A.I. Kozhanov, O.I. Bzheumikhova, Elliptic and Parabolic Equations with Involution and Degeneration at Higher Derivatives, Mathematics, 10 (2022), 1-10; https://doi.org/10.3390/math10183325.
  9. [9] D. Przeworska-Rolewicz, Some boundary value problems with transformed argument, Commentarii Mathematici Helvetici, 17 (1974), 451-457.
  10. [10] M.A. Sadybekov, A.A. Dukenbayeva, Direct and inverse problems for the Poisson equation with equality of flows on a part of the boundary, Complex Var. Elliptic Equ., 64 (2019), 777-791; https://doi.org/10.1080/17476933.2018.1517340.
  11. [11] M.A. Sadybekov, A.A. Dukenbayeva, On boundary value problem of the Samarskii-Ionkin type for the Laplace operator in a ball, Complex Var. Elliptic Equ., 67 (2022), 369-383; https://doi.org/10.1080/17476933.2020.1828377.
  12. [12] M.A. Sadybekov, B.T. Torebek, B.Kh. Turmetov, Representation of Green’s function of the Neumann problem for a multidimensional ball, Complex Var. Elliptic Equ., 61 (2016), 104-123; https://doi.org/10.1080/17476933.2015.1064402.
  13. [13] M.A. Sadybekov, B.Kh. Turmetov, On analogues of periodic boundary value problems for the Laplace operator in a ball, Eurasian Math. J., 3 (2012), 143-146.
  14. [14] M.A. Sadybekov, B.Kh. Turmetov, On an analog of periodic boundary value problems for the Poisson equation in the disk, Diff. Equat., 50 (2014), 268-273; https://doi.org/10.1134/S0012266114020153.
  15. [15] B.Kh. Turmetov, Generalization of the Robin Problem for the Laplace Equation, Diff. Equat., 55 (2019), 1134-1142; https://doi.org/10.1134/S0012266119090027.
  16. [16] B. Turmetov, V. Karachik, On eigenfunctions and eigenvalues of a nonlocal laplace operator with multiple involution, Symmetry, 13 (2021), 1-20; https://doi.org/10.3390/sym13101781.
  17. [17] B. Wirth, Green’s function for the Neumann-Poisson problem on n- dimensional balls, The American Mathematical Monthly, 127 (2020), 737- 743; https://doi.org/10.1080/00029890.2020.1790910.
  18. [18] U. Yarka, S. Fedushko, P. Vesely, The Dirichlet problem for the perturbed elliptic equation, Mathematics, 8 (2020), 1-13; https://doi.org/10.3390/math8122108.
  19. [19] N. Yessirkegenov, Spectral properties of the generalized Samarskii Ionkin type problems, Filomat, 32 (2018), 1019-1024; DOI:10.2298/FIL1803019Y.